Question

Statement-1 : Two particles of mass 1 kg and 3 kg move towards each other under their mutual force of attraction. No other force acts on them. When the relative velocity of approach of the two particles is 2 m/s, their centre of mass has a velocity of 0.5 m/s. When the relative velocity of approach becomes 3 m/s, the velocity of the centre of mass is 0.75 m/s.

Statement-2 : The total kinetic energy as seen from ground is $\frac{1}{2}\mu v_{rel}^{2}+\frac{1}{2}mv_c^{2}$ and in absence of external force, total energy remains conserved.

• A. Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
• B. Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
• C. Statement-1 is true, statement-2 is false.
• D. Statement-1 is false, statement-2 is true.

Answer 1

Answer: (D)

Statement-1 is false, statement-2 is true.

Answer 2

Analysis of Statement-1:

The system consists of two particles of masses $m_1 = 1$ kg and $m_2 = 3$ kg. They move towards each other under their mutual force of attraction. Crucially, "No other force acts on them." This means the system of the two particles is isolated.

For an isolated system, the net external force is zero ($F_{ext} = 0$).

According to Newton's second law for a system of particles, the net external force is equal to the product of the total mass ($M = m_1 + m_2$) and the acceleration of the center of mass ($a_c$):

$F_{ext} = M a_c$

Since $F_{ext} = 0$, it implies $a_c = 0$.

If the acceleration of the center of mass is zero, then its velocity ($v_c$) must be constant.

Statement-1 claims that when the relative velocity of approach changes from 2 m/s to 3 m/s, the velocity of the center of mass changes from 0.5 m/s to 0.75 m/s. This contradicts the principle that the velocity of the center of mass of an isolated system remains constant.

Therefore, Statement-1 is false.

Analysis of Statement-2:

The statement has two parts:

1. "The total kinetic energy as seen from ground is $\frac{1}{2}\mu v_{rel}^{2}+\frac{1}{2}mv_c^{2}$"

The total kinetic energy ($K_{total}$) of a system of particles can be expressed as the sum of the kinetic energy of the center of mass ($K_{CM}$) and the kinetic energy of the particles relative to the center of mass ($K_{rel}$).

$K_{total} = K_{CM} + K_{rel}$

For a two-particle system with total mass $M = m_1 + m_2$, the kinetic energy of the center of mass is $K_{CM} = \frac{1}{2} M v_c^2$.

The kinetic energy of the particles relative to the center of mass is $K_{rel} = \frac{1}{2} \mu v_{rel}^2$, where $\mu = \frac{m_1 m_2}{m_1 + m_2}$ is the reduced mass and $v_{rel}$ is the magnitude of the relative velocity.

So, $K_{total} = \frac{1}{2} M v_c^2 + \frac{1}{2} \mu v_{rel}^2$.

The statement uses 'm' instead of 'M' for the total mass. In the context of such formulas, 'm' is typically understood as the total mass $M = m_1 + m_2$. Assuming 'm' refers to the total mass, this part of the statement is correct.

2. "and in absence of external force, total energy remains conserved."

For a system where only internal forces (like mutual gravitational attraction, which is a conservative force) are acting and no external forces are present, the total mechanical energy (sum of kinetic and potential energy) of the system remains conserved. This is a fundamental principle of energy conservation.

Therefore, this part of the statement is also correct.

Combining both parts, Statement-2 is true (under the reasonable assumption that 'm' denotes the total mass).

Conclusion:

Statement-1 is false.

Statement-2 is true.